We have used the Gemini Multi-Object Spectragraph (GMOS) instrument on the 8.1-m Gemini-South Telescope to obtain spatially resolved two-colour imaging and integral field unit (IFU) spectroscopy of a sample of 10 nearby (z = 0.04-0.20) 'E+A' galaxies selected from the Two Degree Field Galaxy Redshift Survey. These galaxies have been selected to lie in a variety of environments from isolated systems to rich clusters. Surface brightness profiles measured using our imaging data show the isophotal profiles of our sample are generally r(1/4)-like, consistent with a sample dominated by early-type galaxies. Only one galaxy in our sample has an obvious exponential ('disc-like') component in the isophotal profile. This is further underscored by all galaxies having early Hubble-type morphological classifications, and showing a behaviour in the central velocity dispersion-absolute magnitude plane that is consistent with the Faber-Jackson relation, once the transitory brightening that occurs in the E+A phase is corrected for. In addition, two-thirds of our sample shows clear evidence of either ongoing or recent tidal interactions/mergers, as evidenced by the presence of tidal tails and disturbed morphologies. While all the galaxies in our sample have total integrated colours that are relatively blue (in keeping with their E+A status), they show a diversity of colour gradients, possessing central core regions that are either redder, bluer or indistinct in colour relative to their outer regions. The E+A spectra are well fitted by that of a young stellar population, the light from which is so dominant that it is impossible to quantify the presence of the underlying old stellar population. Consistent with other recent findings, there is little evidence for radial gradients in the Balmer absorption line equivalent widths over the central few kiloparsecs (< 4 kpc), although we are unable to search for the previously reported radial gradients at larger galactocentric radii due to the limited spatial extent of our IFU data. Kinematically, the most striking property is the significant and unambiguous rotation that is seen in all our E+A galaxies, with it being generally aligned close to the photometric major axis. This is contrary to the findings of Norton et al., who found little or no evidence for rotation in a very similar sample of nearby E+A galaxies. We also clearly demonstrate that our E+A galaxies are, in all but one case, consistent with being 'fast rotators', based on their internal angular momentum per unit mass measured as a function of radius and ellipticity. We argue that the combination of disturbed morphologies and significant rotation in these galaxies supports their production via gas-rich galaxy mergers and interactions. The large fraction of fast rotators argues against equal mass mergers being the dominant progenitor to the E+A population.